{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T16:26:25Z","timestamp":1761582385220,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100018227","name":"National Research Foundation of Ukraine","doi-asserted-by":"publisher","award":["2023.03\/0198"],"award-info":[{"award-number":["2023.03\/0198"]}],"id":[{"id":"10.13039\/100018227","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we introduce \u03c9n-symmetric polynomials associated with the finite group \u03c9n, which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces \u21131. These polynomials extend the classical notions of symmetric and supersymmetric polynomials on \u21131. We explore algebraic bases in the algebra of \u03c9n-symmetric polynomials and derive corresponding generating functions. Building on this foundation, we construct rings of multisets (multinumbers), defined as equivalence classes on the underlying space under the action of \u03c9n-symmetric polynomials, and investigate their fundamental properties. Furthermore, we examine the ring of integer multinumbers associated with the group \u03c9n, proving that it forms an integral domain when n is prime or n=4.<\/jats:p>","DOI":"10.3390\/sym17010033","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:13:32Z","timestamp":1735290812000},"page":"33","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Symmetric Functions and Rings of Multinumbers Associated with Finite Groups"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0594-3214","authenticated-orcid":false,"given":"Yurii","family":"Chopiuk","sequence":"first","affiliation":[{"name":"Computer Science Department, Kyiv School of Economics, 3 Shpaka Str., 03113 Kyiv, Ukraine"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5554-4342","authenticated-orcid":false,"given":"Andriy","family":"Zagorodnyuk","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Liu, X.D., and Pedrycz, W. 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